Lesson 3.5 – Finding the domain of a Rational Function

To find the domain set the denominator to zero and solve for x. The domain will be all real number except that value for x.

The denominator is x- 3.

Solve for x.

x- 3 = 0

x = 3

The domain is all Real Numbers

Except x≠3

2 9

3

x

x

Example 2

Find the domain:

X2 + 9 = 0

X2 = -9

The domain is only for real numbers not imaginary the domain is(-∞, ∞)

2

3

9

x

x

YOU TRY!!!

Find the domain of

Find the domain of

Answers

1. All real numbers except x≠5 or -5

2. All real numbers.

2 25

x

x

2

3

16

x

x

The parent function of rational functions is 1

( )f xx

What does the graph look like?

Another Basic Rational Functionf(x) = 1/x2 and it looks like this:

Asymptotes: lines that a graph approaches but does not cross

Vertical asymptotes: Whichever values are not allowed in the

domain will be vertical asymptotes on the graph.

Where is the domain limited?

Set those factors that only appear in the denominator or those that appear more times in the denominator than numerator equal to zero and solve.

denominator

Example1.Find the vertical asymptotes:

Set x – 2 = 0, x = 2 is a vertical asymptote.

2.Find the vertical asymptotes:

Factor x2 – 9 = (x-3)(x+3)

Set x – 3 = 0, x =3 is the vertical asymptote

There won’t be one at x=-3, which means there is a hole in the graph at -3 or point discontinuity. Picture next slide.

3 1( )

2

xf x

x

9

3)(

2

x

xxf

)3)(3(

3)(

xx

xxf

3

1

x

Horizontal asymptotes:

Look at the degrees of the numerator and denominatorIf the degrees are equal then the horizontal

asymptote is the ratio of the leading coefficients

( y = ratio of leading coefficients)If the degree in the denominator is greater then

the horizontal asymptote is y = 0If the degree in the numerator is greater then

there is no horizontal asymptote.

Definition of Horizontal Asymptotes

Ex 1: Find the horizontal asymptotes of each rational functionA)

Horizontal: degrees are equal (both are 1st

degree) so y = ratio 3/1

y = 3

B)

Horizontal: If the degree in the denominator is greater than the numerator horizontal asymptote is y = 0

3 1( )

2

xf x

x

12

4)(

2 x

xxf

C)

If the degree in the numerator is greater than there is no horizontal asymptote.

12

4)(

2

3

x

xxf

YOU TRY!!! Ex 1: find the vertical & horizontal asymptotes of each rational functionA)

Vertical: x-5=0 x = 5

Horizontal: degrees are equal (both are 1st

degree) so y = ratio 4/1

y = 4

B)

Vertical: x+1 = 0

x+3=0

So x = -1, x = -3 Horizontal: denominator is

bigger (3rd degree vs. 2nd)

so y = 0

x = -2 is a hole

2 3 10( )

( 1)( 2)( 3)

x xf x

x x x

5

14)(

x

xxf

Lesson 3.5 Graphing a Rational Function

Rational Functions that are not transformations of f(x) = 1/x or f(x) = 1/x2 can be graphed using the following suggestions.

Strategy for Graphing a Rational Function

The following strategy can be used to graph f(x) = p(x)

q(x),

Where p and q are polynomial functions with no common factors.

Seven steps:

Step 1: Determine whether the graph of f has symmetry.

f(-x) = f(x): y-axis symmetry

f(-x) = -f(x): origin symmetry

•Step 2: Find the y-intercept (if there is one) by evaluating f(0).

•Step 3: Find the x-intercepts (if there are any) by solving the equation p(x) = 0.

Steps continued:

Step 4: Find any vertical asymptote(s) by solving

the equation q(x) =0. •Step 5: Find the horizontal asymptote (if there is one ) using the rule for determining the horizontal asymptote of a rational function.

•Step 6: Plot at least one point between and beyond each x-intercept and vertical asymptote.

•Step 7: Use the information obtained previously to graph the function between and beyond the vertical asymptotes.

Example 1: Graph: f(x) = 2x x-1

Example 2: 2

2

3( )

4

xf x

x

Step 1: Determine Symmetry: f(-x)

Step 2: Find the Y-intercept. f(0)

Step 3: Find the x-intercepts. p(x) = 0

Step 4: Find the vertical asymptote(s). q(x)=0

Step 5: Find the horizontal asymptote. (Degree of numerator and denominator)

Step 6: Plot points between and beyond each x-intercept and vertical asymptote. (table)

Step 7: Graph the function.

2

2

3( )

4

xf x

x

2 2

2 2

3( ) 31. ( )

( ) 4 4

x xf x

x x

The graph of f is symmetric with respect to the y-axis.

2. f(0) = 3*02 = 0 = 0

02 – 4 -4

The y-intercept is 0, so the graph passes thru the origin.

3. 3x2=0 , so x = 0. The x-intercepts is 0, verifying the graph passes through the origin.

4. Set q(x) =0,

x2-4 = 0, so x = 2 and x= -2

The vertical asymptotes are x = -2 and x=2.

Ex. 2 cont:

5. Look at the degree of numerator and denominator. They are equal so you use the leading coefficients. 3/1

The horizontal asymptote is y = 3.

6. Plot points between and beyond each x-intercept and vertical asymptote.

X -3 -1 1 3 4

f(x) 3x2

x2 -4

27/5

-1 -1 27/5 4

7. Graph the functions.

Summary:

If you are given the equation of a rational function, explain how to find the vertical asymptotes of the function.